\[ \int \frac {\left (4+x^5\right ) \sqrt {1-2 x^5+x^8+x^{10}}}{x^9} \, dx \]
Optimal antiderivative \[ \frac {\left (x^{5}-1\right ) \sqrt {x^{10}+x^{8}-2 x^{5}+1}}{2 x^{8}}-2 \ln \left (x \right )+\frac {\ln \left (-1+x^{5}+\sqrt {x^{10}+x^{8}-2 x^{5}+1}\right )}{2} \]
command
Integrate[((4 + x^5)*Sqrt[1 - 2*x^5 + x^8 + x^10])/x^9,x]
Mathematica 13.1 output
\[ \frac {\left (-1+x^5\right ) \sqrt {1-2 x^5+x^8+x^{10}}}{2 x^8}-\frac {1}{2} \tanh ^{-1}\left (\frac {1-x^5}{\sqrt {1-2 x^5+x^8+x^{10}}}\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (4+x^5\right ) \sqrt {1-2 x^5+x^8+x^{10}}}{x^9} \, dx \]________________________________________________________________________________________